epl draft
arXiv:1008.0267v1 [cond-mat.stat-mech] 2 Aug 2010
Enumeration of spanning trees in a pseudofractal scale-free web Zhongzhi Zhang1,2 1 2
(a)
, Hongxiao Liu1,2 , Bin Wu1,2 and Shuigeng Zhou1,2
(b)
School of Computer Science, Fudan University, Shanghai 200433, China Shanghai Key Lab of Intelligent Information Processing, Fudan University, Shanghai 200433, China
PACS PACS PACS
89.75.Hc – Networks and genealogical trees 05.50.+q – Lattice theory and statistics (Ising, Potts, etc.) 05.20.-y – Classical statistical mechanics
Abstract. - Spanning trees are an important quantity characterizing the reliability of a network, however, explicitly determining the number of spanning trees in networks is a theoretical challenge. In this paper, we study the number of spanning trees in a small-world scale-free network and obtain the exact expressions. We find that the entropy of spanning trees in the studied network is less than 1, which is in sharp contrast to previous result for the regular lattice with the same average degree, the entropy of which is higher than 1. Thus, the number of spanning trees in the scale-free network is much less than that of the corresponding regular lattice. We present that this difference lies in disparate structure of the two networks. Since scale-free networks are more robust than regular networks under random attack, our result can lead to the counterintuitive conclusion that a network with more spanning trees may be relatively unreliable.
Introduction. – The enumeration of spanning trees in networks (graphs) is a fundamental issue in mathematics [1–3], physics [4,5], and other discipline [6]. A spanning tree of any connected network is defined as a minimal set of edges that connect every node. The problem of spanning trees is relevant to various aspects of networks, such as reliability [7, 8], optimal synchronization [9], standard random walks [10], and loop-erased random walks [11]. In particular, the number of spanning trees corresponds to the partition function of the q-state Potts model [12] in the limit of q approaching zero, which in turn closely relates to the sandpile model [13]. Because of the diverse applications in a number of fields [14], a lot of efforts have been devoted to the study of spanning trees. For example, the exact number of spanning trees in regular lattices [4, 15] and Sierpinski gaskets [5] has been explicitly determined in previous studies. However, regular lattices and fractals cannot well mimic the real-life networks, which have been recently found to synchronously exhibit two striking properties: scalefree behavior [16] and small-world effects [17] that has a strong impact on the enumeration problems on networks. For example, previous work on counting subgraphs, such as cliques [18], loops and Hamiltonian cycles [19], has (a)
[email protected] (b)
[email protected]
shown that scale-free degree distribution implies a very non-trivial structure of subgraphs. However, so far investigation on the number of spanning trees in scale-free small-world networks has been still missing. In view of the distinct structure, as compared to regular lattices, it is of great interest to examine spanning trees in scale-free small-world networks. In this paper, we intend to fill this gap by providing a first analytical research of spanning trees in a smallworld network with inhomogeneous connectivity. In order to exactly obtain the number of spanning trees, by using a renormalization group method [20], we consider a deterministically growing scale-free network with small-world effect. We find that the entropy of its spanning trees is smaller than 1, which is a striking result that is qualitatively different from that of two dimensional regular lattices with identical average degree, in which the entropy is higher than 1. Thus, the number of its spanning trees is much lower than that of its corresponding regular lattice. We show that this difference can be accounted for by the heterogeneous structure of scale-free networks. Since the network under study is much robust to random deletion of edges, as opposed to regular lattice, our result suggests that networks with more spanning trees are not always more stable to random breakdown of edges, compared with those networks with less spanning trees.
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Zhongzhi Zhang, Hongxiao Liu, Bin Wu, Shuigeng Zhou
Fig. 1: The first three generations of the iterative scale-free network.
Pseudofractal scale-free web. – The studied scalefree network [21, 22], denoted by Gn after n (n ≥ 0) generations, is constructed as follows: For n = 0, G0 is a triangle. For n ≥ 1, Gn is obtained from Gn−1 : every existing edge in Gn−1 introduces a new node connected to both ends of the edge. Figure 1 illustrates the construction process for the first three generations. The network exhibits some typical properties of real networks. Its degree distribution P (k) obeys a power law P (k) ∼ k 1+ln 3/ ln 2 , the average distance scales logarithmically with network order (number of nodes) [23], and the clustering coefficient is 45 . Alternatively, the network can be also created in another method [23, 24]. Given the generation n, Gn+1 may be obtained by joining at the hubs (the most connected nodes) three copies of Gn , see Fig. 2. According to the latter construction algorithm, we can easily compute the n+1 network order of Gn is Vn = 3 2 +3 . In Gn , there are three hubs denoted by An , Bn , and Cn , respectively. Number of spanning trees. – After introducing the network construction and its properties, next we will study both numerically and analytically spanning trees in this scale-free network.
Fig. 2: (Color online) Second construction method of the network. Gn+1 can be obtained by joining three copies of Gn (η) denoted as Gn (η = 1, 2, 3), the three hubs of which are rep(η) (η) (η) resented by An , Bn , and Cn . In the merging process, hubs (3) (2) (3) (2) (1) (1) An (resp. Cn , An ) and Bn (resp. Bn , Cn ) are identified as a hub node An+1 (resp. Bn+1 , Cn+1 ) in Gn+1 .
which is a finite number and a very interesting quantity characterizing the network structure. It should be mentioned that although the expression of Eq. (1) seems compact, the computation of eigenvalues of a matrix of order Vn × Vn makes heavy demands on time and computational resources for large networks. Thus, one can count the number of spanning trees by directly calculating the eigenvalues only for the first several iterations, which is not acceptable for large graphs. Particularly, by using the eigenvalue method it is difficult and even impossible to obtain the entropy EGn . It is thus of significant practical importance to develop a computationally cheaper method for enumerating spanning trees that is devoid of calculating eigenvalues. Fortunately, the iterative network construction permits to calculate recursively NST (n) and EGn to obtain exact solutions.
Closed-form formula. To get around the difficulties of the eigenvalue method, we use an analytic technique based on a decimation procedure [20]. For simplicity, we use tn to express NST (n). Moreover, let an denote the number Numerical solution. According to the well-known re- of spanning subgraphs of Gn consisting of two trees such sult [25], we can obtain numerically but exactly the num- that the hub node An belongs to one tree and the two ber of spanning trees, NST (n), by computing the non-zero other hubs (Bn and Cn ) are in the other tree. Analoeigenvalues of the Laplacian matrix associated with Gn as gously, we can define quantities bn and cn , see Fig. 4. By symmetry, we have an = bn = cn . Thus, in the following i=V n −1 Y computation, we will replace bn and cn by an . 1 λi (n) , (1) NST (n) = Considering the self-similar network structure, the folVn i=1 lowing fundamental relations can be established: where λi (n) (i = 1, 2, . . . , Vn − 1) are the Vn − 1 nonzero eigenvalues of the Laplacian matrix for Gn . For a network, the non-diagonal element lij (i 6= j) of its Laplacian matrix is -1 (or 0) if nodes i and j are (or not) directly connected, while the diagonal entry lii equals the degree of node i. Using Eq. (1), we can calculate directly the number of spanning trees NST (n) of Gn (see Fig. 3). From Fig. 3, we can see that NST (n) approximately grows exponentially in Vn . This allows to define the entropy of spanning trees for Gn as the limiting value [1–3] EGn = lim
Vn →∞
ln NST (n) , Vn
(2)
tn+1 = (tn )2 (an + cn + an + bn + cn + bn ) = 6an (tn )2 (3) and an+1 = tn [(cn )2 + an bn + bn cn + an cn ] = 4tn (an )2 . (4) Equation (3) can be explained as follows. Since Gn+1 is obtained via merging three Gn by identifying three couples of hub nodes, to get the number of spanning trees tn+1 for Gn+1 , one of the copies of Gn must be spanned by two trees. There are six possibilities as shown in Fig. 5, from which it is easy to derive Eq. (3). Analogously, Eq. (4) can be understood based on Fig. 6.
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Enumeration of spanning trees in a pseudofractal scale-free web 10000
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Fig. 4: Illustrative definition for the spanning subgraphs of Gn . The two hub nodes connected by a solid line are in one tree, and the two hub nodes linked by a dotted line belong to different trees.
10 Analytical results Numerical results
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Fig. 3: Logarithm of the number of spanning trees NST (n) in network Gn as a function of network order Vn on a log-log scale. In the figure, Ln = ln NST (n); the filled symbols are the numerical results obtained from Eq. (1), while the empty symbols correspond to the exact values from Eq. (11), both of which completely agree with each other.
full range of 0 ≤ n ≤ 8, they are perfectly consistent with each other, which shows that the analytical formulas provided by Eqs. (9) and (11) are right. Figure 3 shows the comparison between the numerical and analytical results. Equation (11) unveils the explicit dependence relation of NST (n) on the network order Vn . Inserting Eq. (11) into Eq. (2), it is easy to obtain the entropy of spanning trees for Gn given by
To obtain tn , we define an intermediary variable hn = tn an that obeys the following recursive relation
EGn = lim
Vn →∞
1 ln NST (n) = (ln 2 + ln 3) ≃ 0.89588 . (12) Vn 2
This obtained asymptotic value is the smallest entropy hn+1 (5) (lower than 1) that has not been reported earlier for other networks with an average degree of 4. For example, the enWith the initial condition t0 = 3 and a0 = 1, we have tropy for spanning trees in the square lattice is 1.16624 [4], a value larger than 1. Thus, the number of spanning trees h0 = 3. Hence, Eq. (5) is solved to yield in Gn is much less than that in the square lattice with the n+1 same average degree of nodes. 3 (6) hn = n . From the result obtained above, we can conclude that 2 the pseudofractal scale-free network has less spanning Then, trees than the regular lattice with the same average de2n (7) gree. The difference can be attributed to the structural an = n+1 tn . 3 characteristics of the two classes of networks. In scalePlugging this expression into Eq. (3) leads to free networks, nodes have a heterogeneous connectivity, which leads to an inhomogeneous distribution of Lapla2n+1 3 (8) cian spectra [21, 26, 27]. On the contrary, in regular lattn+1 = n (tn ) . 3 tices, since all nodes have approximately the same degree, Considering the initial value t0 = 3, we can solve Eq. (8) their Laplacian spectra have a homogenous distribution. Thus, for two given scale-free and regular networks with to obtain the explicit solution the same order and average node degree, the sum of the n+1 n+1 NST (n) = tn = 2(3 −2n−3)/4 3(3 +2n+1)/4 . (9) eigenvalues of their Laplacian matrices are the same, but the product of non-zero Laplacian spectra of the scaleAnalogously, we can derive the exact formula for an as: free network is smaller than its counterpart of the regular network, because of the different distributions of the n+1 n+1 an = 2(3 +2n−3)/4 3(3 −2n−3)/4 . (10) Laplacian spectra resulting from their distinct connectivIt not difficult to represent NST (n) as a function of the ity distribution. Hence, the heterogeneous structure is network order Vn , with the aim to obtain the relation be- responsible for the difference of number of spanning trees n+1 in scale-free networks and regular lattices. It should be tween the two quantities. Recalling Vn = 3 2 +3 , we have stressed that although we only study a specific determin3n+1 = 2Vn − 3 and n + 1 = ln(2Vn − 3)/ ln 3. These istic scale-free network, we expect to find a qualitatively relations enable one to write NST (n) in terms of Vn as similar result about spanning trees in real-world scale-free [Vn −ln(2Vn −3)/ ln 3−2]/2 [Vn +ln(2Vn −3)/ ln 3−2]/2 networks, since they have similar structural characteristics NST (n) = 2 3 . (11) as that discussed above. We have confirmed the closed-form expressions for As an important invariant of a network, the number of NST (n) against direct computation from Eq. (1). In the spanning trees is a relevant measure of the reliability of the tn+1 3tn 3 = = = hn . an+1 2an 2
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Zhongzhi Zhang, Hongxiao Liu, Bin Wu, Shuigeng Zhou
Fig. 5: (Color online) Illustration for the recursion expression for the number of spanning trees tn+1 in Gn+1 . The two nodes at both ends of a solid line are in one tree, while the two nodes at both ends of a dotted line are in separate trees. Fig. 6: (Color online) Illustration for the recursive expression for the number of spanning subgraphs an+1 corresponding to network Gn+1 . The two nodes at both ends of a solid line (dotted line) are in one tree (two trees).
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